Application of fractional-order nonlinear equations in coordinated control of multi-agent systems

: In order to solve the coordinated operation of voltage and frequency of microgrids and achieve e ﬀ ective distribution of output power, we propose the application of fractional-order nonlinear equations in coordination. This method designs a distributed impulse coordinated control strategy, to achieve the coordinated operation of the system. The distributed coordinated control structure and mathematical model are established, and the distributed two-level coordinated control strategy of the microgrid system is adopted. Aiming at the secondary control problem of micro-grids, a distributed coordinated control protocol is designed. The results showed that after adding the distributed second-level coordination control of frequency at 3 s, the output active power of the four distributed power sources in the microgrid model is maintained to an evenly divided state after about 1 s. The output voltage and frequency utilizing the microgrid ’ s decentralized power supply quickly reach the ideal reference value, within the allowable error range, the output of the system can achieve coordinated control, and the active power can be distributed proportionally.


Introduction
The emergence of multi-agent systems provides an effective method for studying tasks that cannot be completed by a single individual.Multi-agent systems belong to the distributed direction of artificial intelligence; therefore, with the continuous in-depth research and wide application of artificial intelligence, the distributed coordinated control problem of multi-agent systems has attracted the attention of many domestic and foreign scholars, and they have devoted their efforts and has achieved good research results in different fields, such as physics, science, and mathematics.
In these fields, the coordinated study of multi-agent systems has played a very important role in the information communication between each individual in the system and its neighbors.The corresponding control strategy is applied according to the corresponding control objective, so that each individual in the system forms a cooperative relationship with each other, and the entire system completes difficult target tasks.Such a collaborative working mechanism effectively solves the shortcomings of centralized information processing; at the same time, each agent can be effectively used, and the limited ability of the individual can be broken through.In the coordinated control of the system, the agent can design the relevant controller only by obtaining the information of its directly connected neighbors.This greatly reduces the cost of information transmission between agents, so the research on this coordinated control strategy is of great significance.
The great significance of the research on the stability frequency of the microgrid system in real life is combined with the reference values.If there is a large difference between the dispersed power supply's electrical output voltage and duration, as well as each of their corresponding reference values, this will make the system unable to operate stably, affect the transmission and use of electric energy, and cause huge economic and property losses.The significant importance of the study of the equilibrium of the microgrid system's output voltage and output frequency in real-world applications is combined with the reference values.Therefore, it is necessary to select an appropriate control protocol, and the frequency of the system can be stabilized near the reference value in the islanding operation mode, ensuring the stable operation of the microgrid system.In the microgrid, the main purpose of the secondary control in its hierarchical control structure is to ensure that the output voltage and output frequency of the system return to their respective reference values; therefore, it is of great value to study the secondary control of the microgrid system (Figure 1).

Literature review
Aghayan et al. proposed and studied the coordinated control for the first time, and the disseminated coordinated controller problem of structures was also tackled.Complex networks with directed connection graphs were considered; at the same time, the formation in the case of dynamic connection and absolute/relative impedance were also considered [1].This study mainly focuses on different orders of multi-agent systems, linear or nonlinear, continuous or discrete, different network topologies, state observability, whether there is time delay, etc.In the research on the control strategy of the coordinated control of the fractional order system, in addition to the complex network, the study of several agents has also obtained many achievements.Bertsekas considered fractional-order multi-agent systems with nonlinear properties; its coordinated control was studied for its leader-follower model, where the feedback gain of the controller was easily determined by linear equations or inequalities [2].Bingqiang et al. discussed the fractional-order multi-agent system with reference states, and the introduction of reference states plays a significant part in the study of synchronized system administration [3].Anjum et al., for the system of leader-follower motion model, combined the stability analysis method of the fractional order system.An adaptive pinning control method was proposed to realize the coordinated control of the system.In that method, self-adaptation was used in the processing of nonlinear systems, and the introduction of pinning control reduced the number of controllers in the system and greatly reduced the control cost [4].Liang et al. proposed a generalized distributed two-level coordination control strategy, but it required information transmission between each individual and all other individuals, and its communication cost was not much different from the centralized one [5].Dantus et al. treated the second-order frequency control problem as linear, and therefore the problem was reduced to conventional system analysis [6].The cost of information transmission between agents was significantly reduced when the system was controlled in a coordinated manner because an agent can only design the appropriate controller by using information from its directly connected neighbors.For this reason, research on this coordinated control strategy is very important, combined with significant importance to the study of the stability frequency of the microgrid system in practical application to the reference values.We conducted in-depth research and analysis on the coordinated control and discussed and designed the distributed impulse control method to realize the stable state of the system and synchronous motion.Considering the application of the multi-agent coordination problem, distributed control technology is introduced into the synchronous operation of voltage and power in microgrids.The distributed control structure is adopted since the information redundancy generated in the communication of the system is reduced.This will greatly improve the accuracy of the information used by the system and increase the credibility of the system.And for the layered structure of the microgrid, the control strategy of fractional-order PID is selected in the voltage and current loops, which makes the stability of the system superior.

Lyapunov stability theory
Equilibrium point: If the system has a state point x e in the dynamic process, each state component in the system maintains a balance and no longer changes with time, that is, The state point x e is called the equilibrium state of the dynamic system.
Lyapunov stable: Knowing that δ > 0 and ε > 0, when time t tends to infinity, the state trajectory of the system starting from the initial state ( ) ∈ x S δ 0 will never exceed the closed-sphere region ( ) S ε , that is, the state amplitude of the dynamic response of the system is always a bounded value, and the system is called Lyapunov-stable here [7].
Asymptotically stable: it is known that δ > 0 and ε > 0; assuming that there is an equilibrium state that satisfies Lyapunov stability, and the system state trajectory starting from ( ) S δ will not exceed the range ( ) S ε of the closed sphere anyway, and the state will converge to the equilibrium state with time, then the system is asymptotically stable in the equilibrium state.
Consistently stable: Based on the above definition of asymptotically stable, if the range of the closed-sphere area ( ) S δ has nothing to do with the initial time t 0 of the system, the equilibrium state is said to be uniformly asymptotically stable.
Lyapunov direct method: Design the energy function V (x, t) of the system; call this function the Lyapunov function, find the rate of change of energy with time V(x, t), and observe the alteration rule of the energy function conferring to the state equation of the organization; finally, the positive definite characteristics of V(x, t) and V(x, t) are used to judge the stability at the equilibrium state.

Pulse control theory
Based on the mathematical knowledge of impulse differential equations, impulse control methods have been studied in-depth.In the case of applying impulse control, at least one state variable is included, which can change instantaneously at least at one time.The following will introduce the definition of pulse control and several characteristics of using this method.
Assuming that P is a continuous control system, its state variable , the input expressed as ( ) n , when = t τ k , the state variable x has a sudden change of pulse, that is, ( ) m is output under the action of this control input, when k → ∞ (that is, time t → ∞), when the system output is infinitely close to the reference value, that is, → ∈ * y y R m , the control method is called pulse control [8].
In the above definition, the control system P is in the non-impulsive form, while the control law is in the impulse form.But the complete impulse control system also includes that the control system P is in the form of impulse, and the control law is in the form of non-impulse.Accordingly, it can be divided into the following two types (2).The first sort: where x and y are the state variables and output variables of the system, and is the impulse control law.The second type of formula (3): In this type of impulse control system, there are two control inputs, namely the impulse control law ( ) U k y , and the continuous control law u ¯.
The research on pulse control is mainly the first type of pulse control system, and the designed control law is in the form of pulse; therefore, it is necessary to study the pulse interval of the pulse controller and the selection conditions of the pulse gain, and achieve the control objectives of the system and run stably.

Definition of fractional calculus
In fractional calculus, its expression is relatively uniform, and it is represented by an operation operator, which is recorded as , where α and t, respectively, represent the upper and lower limits of the calculus operation, and α represents the order in the calculus operation [9].Therefore, the operation of fractional order is as follows.
Due to the short development time of fractional calculus, its definition has not been uniformly identified.The main issues discussed at present are generally based on the following three definitions (4): ① Grinwald-Letnikov definition: where ! ,α is any real number, h is the calculation step size, and [ ] ⋅ is the rounding operation.
② Riemann-Liouville definition: For any real number , the differential is defined as in the Eq. ( 5): The integral is defined as In the above two formulae, ( ) ⋅ Γ is the gamma function, and its expression is as shown in formula (7): In formula (7), the value of x is that the real part is greater than zero, that is, Re(x) > 0. The function satisfies the condition ( ) ( ) . Its calculus definition can be written uniformly as formula ( 8): ③ Caputo definition ( 9): where 3.2 Impulsive coordinated control of fractional-order multi-agent systems

Problem description
In order to study the problem discussed in this section, a fractional-order multi-agent system with N nodes of the same structure and nonlinear dynamics is reflected, and its state equation can be described as in Eq. ( 10): where A R n n is a constant matrix, and ( ) u t i represents the control input of node i.In this section, all agents share a state-space vector R n , that is, the state vector of each agent is denoted as . The dynamic model of the leader is considered as in formula (11): where x 0 (t) ∈ R n is the leader's state contingent, which is comparable to an external command generator or system that can produce the desired target trajectories.We define In this section, in order to make the follower effectively track the given leader goal, it is assumed that the agents that have a direct connection with each other can communicate with each other at each discrete impulse control time.Then, the impulse controller input of the ith agent with distributed structure can be described as in Eq. ( 12): where 1, 2, ... , , k n n is the nonlinear dynamic control gain, and ( ) * δ t is the impulse control func- tion of ( ) = * δ t 0 that satisfies the condition when t ≠ 0. The discrete time sequence} satisfies the following conditions: From this expression, it can be seen that the controller only needs to obtain its state information and its neighbors at discrete points in time to realize the input information, which greatly reduces the cost of communication [10].

Controller design
If there exists ξ > 1 such that the following inequality holds (13): where parameters λ λ λ , , ˜˜, T 0 0 , respectively.Then, the dynamic systems with multiple agents with fractional order (10) can realize coordinated control under the action of the distributed impulse controller (12) [11].
Proof.Select the Lyapunov function V(t), assuming Eq. ( 14): On the trajectory of this equation, find the fractional derivative of V(t), when , and the following inequality relation (15) can be obtained: According to the above analysis of a certain period of time and a specific moment, in the ( ] ∈ t t t , 0 1 time, the Lyapunov function can be written as From this, V(t) can be calculated when t = t 1 as At the same time, formula (17) of V(t) at the moment after the system state jumps at t = t 1 can be obtained: According to the same analysis method above, using the recursive idea, we can obtain formula (18) when According to formula (24), formula ( 19) can be obtained: According to (15), formula (20) can be obtained: Because , ( ) V t 0 are all finite constants, and ξ is a constant greater than 1, so when k → ∞ (that is, t → ∞), → ξ 1/ 0 k .Therefore, it can be easily deduced that the error vector δ converges to zero globally, that is, the state trajectories of all follower nodes x i and the state trajectories of leader node x 0 finally achieve global synchronization, so that the states of all agents in the system are coordinated.The first type of pulse control system is the focus of most pulse control research.Because the designed control law takes the form of a pulse, it is important to understand the pulse interval of the pulse controller and the conditions under which the pulse gain should be chosen in order to meet the system's control goals and maintain stability.This completes the proof.

Distributed two-level coordinated
control of the microgrid

Distributed power modeling
The block diagram of the fundamental design of a distributed power supply based on a voltage source inverter is shown in Figure 2. The block diagram includes three control loops, which are control loops for power, voltage, and current, respectively; the structure also includes a LC filter and an output interface [12].
If the operating frequency of the ith distributed generation unit in the microgrid is ω i , and there is a unit in the system whose operating frequency is ω com , the operating frequency value is taken as the reference output value of the entire microgrid.Therefore, taking this frequency as the reference origin, the operating frequencies corresponding to other units can be represented as ϖ i and satisfy the fol- lowing differential equation (Eq.( 21)): It can be seen from the structural block diagram of Figure 2 that the power control loop acts through the droop control.The reference input values * v odi and * v oqi and the operating frequency ω i required by the system are pro- vided for the voltage control loop and the inverter, respectively [13].The active and reactive power values at each moment can be calculated by the measured output voltage and output current value, and by the action a filter known as low-pass ω ci , with a cut-off frequency range, the system output active and reactive power can be obtained.The fundamental components of power, denoted P i and Q i , respectively, are obtained as: In the above equations, v i v i , , , odi odi oqi oqi are the dand q-axis components of v oi and i oi in Figure 2, respectively.
Under the action of the power control loop, the relationship between the output of the average distribution power's frequencies source and the vigorous control, energy, and reactive power can be expressed as formula (24): The parameters ω ni and V ni , respectively, represent the reference value of the output frequency and voltage of the ith distributed unit, and the parameter m Pi and the para- meter n Qi represent the droop control coefficient of the output regularity and voltage, respectively.
The current controller uses the reference voltages * v odi and * v oqi provided by the upper-level control, and provides reference values * i ldi and * i lqi for the lower-level current control through the action of the fractional-order PI D λ μ controller.Figure 3 displays the block structure.Its corresponding dynamic equation can be expressed as Eqs.( 25)-( 28): In the above formulae, the introduced auxiliary parameters ϕ di and ϕ qi are the outputs of the fractional-order PI D λ μ controller in Figure 3, F i is the feedforward gain, ω b is the reference angular frequency, and K PVi , K IVi , K DVi , λ, and μ are five parameters that need to be adjusted.i ldi and * i lqi and the actual value as the input signal to realize the control function of the system.
Its corresponding state equation is shown in Eqs. ( 29)-( 32): In the above formulae, the introduced auxiliary parameters ψ di and ψ qi are the outputs of the fractional-order PI D λ μ controller in Figure 4, i ldi and i lqi are the d-q quad- rature components of the current i li , respectively, and K PCi , K ICi , K DCi , α, and β are the five parameters that need to be adjusted in the system.In the whole system mathematical model, there are also LC filters and output interfaces, and their differential equations can be expressed as in Eqs. ( 33)-( 38):  Coordinated control of multi-agent systems  7 By calculation and analysis, it is possible to create the mathematical model of the ith power generation unit, which can be written in the following nonlinear equation form: The state vector x i is expressed as formula (40): In the design of the fractional-order PI D λ μ controller of the voltage-current loop, five control parameters need to be set since the genetic algorithm can realize on-line tuning; we adopt the method given in the study of Schmitt and Ulbrich [14].

Synchronized distribution
The synchronized distribution supplementary voltage regulation y i is differentiated twice, and the formula for the direct relationship between y i and u i may be found in formula (41): i i i and L h Fi i are the Lie derivatives of the function h i under the F i trajectory.
The auxiliary variable n i is defined as formula (42): The output dynamics can be written as a subsequent second-degree linear equation: The value of N represents the number of distributed generation units.At this time, the controller design of the cooperative control of the system can be regarded as selecting the appropriate η i to make the output y i of the system achieve synchronization.At this time, the control input of the system is expressed as Eq. ( 43): The communication connection between distributed generation units can be regarded as a directed graph G, and the communication weight between the ith generation unit and the reference value (that is, the virtual leader) is denoted as c i .If the ith distributed generation unit is directly connected to the virtual leader, then c i = 1, otherwise c i = 0. Therefore, the voltage control strategy of the secondary control layer can be expressed as designing an appropriate η i so that r i and r 0 are synchronized, that is, r i → > r 0 .
The local synchronization error of the ith distributed generation unit can be expressed as Eq. ( 44): The control structure of the voltage can be represented as shown in Figure 5.

Distributed secondary coordination control of frequency
In the secondary control strategy of frequency, the effective way to achieve the goal is to choose the appropriate form of frequency control input ω ni .The output frequency amplitude of each distributed generation unit finally approaches the reference value ω ref , and the output active power of each distributed generation unit is distributed according to the corresponding proportion.The output and input of this control are = y ω i i and = u ω i n i , respectively.The control output can be written as Eq. ( 45): Differentiating the above formula, we can obtain formula (46): Let the auxiliary controller be = u ωω i i ; therefore, the following first-order form can be used to express the aforementioned Eq. ( 47): In the designed frequency distributed secondary control, each distributed generation unit can communicate to the directed or undirected path of the communication topology, and the value of the auxiliary controller u ωi is only related to itself and its directly connected neighbor information [15].Consider the distributed local state variable feedback control of the ith unit as the following form (48): where c ω is the feedback coefficient, e ωi is the local neighbor state error, and its expression is Eq. ( 49): The parameter a ij > 0 is the communication weight in the knowledge of graph theory: when c i > 0, it means that the ith unit has a direct connection relationship with the reference value ω ref , otherwise c i = 0.
The above formula is written in the form of global variables and expressed as formula (50): where e e e , ,... , The Lyapunov function is selected as formula (51): where Taking the differential operation on both sides of the above function, the following Eq.( 52) can be obtained: ) ) < 0, V ̇< 0 can be obtained.According to the Lyapunov stability theory, the state equilibrium point of the error system is at the origin, and the equilibrium point is stable, that is, the error vector e ω of the system [16].Gradually approaching the zero point, that is to say, when t → ∞, → ω ω i ref , the output frequency value of each distributed unit is synchronized with the given reference value.
In order to effectively utilize resources and avoid wasting electric energy, the load power needs to be distributed proportionally, and the distribution is carried out according to the droop coefficient [17].Therefore, its output active power needs to satisfy the following continuous equation (Eq.( 53)): = =⋯= m P m P m P .
Suppose the auxiliary controller is = u m P Ṗi Pi i ; then, by differentiating each of the above components, Eq. ( 54) is obtained: In order to realize the coordination mechanism, in the designed distributed coordination controller, each distributed power source can communicate according to the communication topology, and the auxiliary controller u Pi is only related to itself and its neighbor informa- tion.Consider the distributed local state variable feedback control input of the ith unit as the following form (55)-(59): where c p is the feedback coefficient, e pi is the local state error, and its expression is where a ij > 0 is the communication weight.So the auxiliary controller u Pi is Therefore, by integrating the above formula, the expression of the control input ω ni can be obtained, and the input of the whole system is also known as The size of the two parameters c ω and c P in the above formula is directly related to the frequency cooperative response speed of the system [18].

Results and discussion
This research is carried out in the island mode operation, so it is necessary to set the reference values.The corresponding voltage and angular frequency of the electrical system is Hz).In the setting of limitations in the Simulink block diagram, the limitations of the power controller, the voltage and current controller, and the system are shown in Table 1.The parameters are, respectively, = ξ 4, S = 0.01, and { } = W P diag 5,000, 1 , from which the feedback gain matrix K can be calculated as K = [2236 37.6], c ω = 1, and c P = 1.
In the simulation, when t = 0 s, it is assumed that the system is disconnected from the large power grid and enters the island mode operation.Within 0 < t < 3 s, the system only uses the primary control strategy.During this process, although the output voltage and output frequency of its four distributed power sources can be stabilized around a value, at this time, the stable amplitude did not reach the output value of the reference value, and both were smaller than the reference values v ref and ω ref [19].For the purpose of regulating the integrated power supply's values output voltage and output frequency to be near the ideal reference value, at time t = 3 s, the distributed two-level coordination control strategy designed above is added.Figure 6 illustrates the changes in the productivity power amplitude and output frequency amplitude of the four distributed power sources before and after adding the distributed secondary coordinated control, the current loop proposed here adopts the fractional-order PI D λ μ controller and distributed two-level coordinated control algorithm.The research on this synchronized control strategy is very important because it allows an agent to design the appropriate controller only by using information from its directly connected other residents.This greatly reduces the cost of information transmission between agents.The significant importance of the study of the equilibrium of the microgrid system's output voltage and output frequency in real-world applications is combined with the reference values.The four distributed power systems' output frequencies and voltage sources in this system are restored to their respective ideal reference values, and their errors are within a reliable range.Figure 7 shows the power distribution of the four distributed power sources in this simulation example, and it can be seen from the figure that at 3 s, after adding the distributed secondary coordination control of frequency, after about 1 s, the output active power of the four distributed power sources in the microgrid model is maintained to an evenly divided state, even if the equation = m P m P Pi i Pj j is established.This is demonstrated by the layout of the fractionalorder PI D λ μ controller.The function of the distributed second-level DAC and the function of the voltage-current loops coordination controller and the output voltage and frequency utilizing the microgrid's dispersed energy source quickly reach the ideal reference value.Within the allowable range of error, the output of the system is realized to achieve coordinated control, and the proportional distribution of active power is realized [20].

Conclusion
Controlling fractional-order nonlinear multi-agent systems via collaborative coordination is explored using the management capability of impulsive controllers.In particular, using fractional-order nonlinear formulas in the integrated management of multi-agent networks greatly reduces the redundancy of communication information and improves the consistency of the organization.Based on the multiagent system, for the hierarchical control structure of the microgrid system, the fractional-order PI D λ μ controller process is adopted in the voltage and current control loop, and a two-level control algorithm with distributed structure is designed to realize the coordinated and synchronous operation with respect to the microgrid's frequencies and voltages.The pulse coordinated control algorithm proposed in this part, as long as each agent in the system obtains its knowledge as well as that of its neighbors directly communicating with it at the pulse moment and responds, all the controlled agents can track the desired target trajectory.This significantly lowers the overall system management cost.The distributed two-level coordinated control algorithm is studied for the input of voltage and frequency, respectively.Each DG in the algorithm only needs to obtain neighbor DGs that communicate directly with it and respond.In this way, the dispersed system's voltage and frequency variables generation units in the whole system can reach the reference value and coordinated control, which ensures that the system can run stably under ideal conditions even in the island mode.

Figure 2 :
Figure 2: Basic block diagram of distributed power supply based on the voltage source inverter.

Figure 4
Figure 4 displays the fundamental block diagram for the current system.This part uses the difference between the reference values * i ldi and * i lqi and the actual value as the input signal to realize the control function of the system.Its corresponding state equation is shown in Eqs.(29)-(32):

Figure 3 :
Figure 3: Block diagram of the voltage controller.

Figure 5 :
Figure 5: Structure diagram of voltage distributed secondary coordination control.
Coordinated control of multi-agent systems  9

Figure 6 :Figure 7 :
Figure 6: Output voltage and angular frequency amplitude of distributed power supply.